In
mathematics the finite Fourier transform may refer to either
another name for
discrete-time Fourier transform (DTFT) of a finite-length series. E.g.,
F.J.Harris (pp. 52–53) describes the finite Fourier transform as a "continuous periodic function" and the
discrete Fourier transform (DFT) as "a set of samples of the finite Fourier transform". In actual implementation, that is not two separate steps; the DFT replaces the DTFT.[A] So
J.Cooley (pp. 77–78) describes the implementation as discrete finite Fourier transform.
^Harris' motivation for the distinction is to distinguish between an odd-length data sequence with the indices which he calls the finite Fourier transform data window, and a sequence on which is the DFT data window.
References
^
George Bachman, Lawrence Narici, and Edward Beckenstein, Fourier and Wavelet Analysis (Springer, 2004), p. 264
Cooley, J.; Lewis, P.; Welch, P. (1969). "The finite Fourier transform". IEEE Trans. Audio Electroacoustics. 17 (2): 77–85.
doi:
10.1109/TAU.1969.1162036.
Further reading
Rabiner, Lawrence R.; Gold, Bernard (1975). Theory and application of digital signal processing. Englewood Cliffs, N.J.: Prentice-Hall. pp 65–67.
ISBN0139141014.